The expectations E[X(1)], E[Z(1)], and E[Y(1)] of the minimum of n independent geometric, modified geometric, or exponential random variables with matching expectations differ. We show how this is accounted for by stochastic variability and how E[X(1)]/E[Y(1)] equals the expected number of ties at the minimum for the geometric random variables. We then introduce the “shifted geometric distribut...

The Grassmann, Taksar, and Heyman (GTH) algorithm for the computation of the stationary distribution of a finite stochastic matrix is shown to apply for the general case when there is a unique stationary distribution. The approach is elementary and matrix based, with probabilistic arguments avoided, to give insight into the essential structural properties. A byproduct is a necessary and suffici...

It is known that various deterministic and stochastic processes such as asymptotically autonomous differential equations or stochastic approximation processes can be analyzed by relating them to an appropriately chosen semiflow. Here, we introduce the notion of a stochastic process X being a weak asymptotic pseudotrajectory for a semiflow 8 and are interested in the limiting behavior of the emp...

In this paper, a novel approach for quantifying the parametric uncertainty associated with a stochastic problem output is presented. As with Monte-Carlo and stochastic collocation methods, only point-wise evaluations of the stochastic output response surface are required allowing the use of legacy deterministic codes and precluding the need for any dedicated stochastic code to solve the uncerta...

Most physical systems are inevitably affected by uncertainties due to natural variabili-ties or incomplete knowledge about their governing laws. To achieve predictive computer simulations of such systems, a major task is, therefore, to study the impact of these uncertainties on response quantities of interest. Within the probabilistic framework, uncertainties may be represented in the form of r...

We develop a systematic information-theoretic framework for quantification and mitigation of error in probabilistic Lagrangian (i.e., path-based) predictions which are obtained from dynamical systems generated by uncertain (Eulerian) vector fields. This work is motivated the desire to improve complex based either on analytically simplified or data-driven models. derive hierarchy general informa...